Towards Precision Measurements of γ : CLEO-c's Pivotal Role
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TOWARDS PRECISION MEASUREMENTS OF γ : CLEO-C’SPIVOTAL ROLE ANDREW S. POWELL
University of Oxford, Denys Wilkinson Building, Oxford, OX1 3RH, UK
Strategies that utilise the interference effects within B ± → DK ± decays hold greatpotential for improving our sensitivity to the CKM angle γ . However, in order toexploit fully this potential, detailed knowledge of the D meson decay structure isrequired. This essential information can be obtained from the quantum correlated ψ (3770) datasets at CLEO-c. Results of such analyses involving the decay modeD → K πππ , and their importance in the context of LHCb, will be presented.
1. Introduction
A means of testing the internal consistency of the Cabbibo-Kaboyashi-Maskawa (CKM) model, whilst simultaneously searching for signatures ofNew Physics, is to perform precision measurements of the angles that com-pose the unitarity triangle: α , β and γ . While β has been measured withextremely high precision at the B-factories (20 . ± . ◦ ) , determination ofthe other two angles currently presents a considerable experimental chal-lenge; most notably γ which is only constrained by direct measurementswith a precision of ∼ ± ◦ . One of the most promising ways of determin-ing the angle γ is through strategies that exploit the interference withinB ± → DK ± decays. a The most straightforward of these strategies con-siders two-body final states of the D meson, however, an abundance ofadditional information can be gained from strategies that consider multi-body final states instead. In order to exploit fully the wealth of statisticssoon to arrive at the LHC, the LHCb , experiment plans to utilise all suchstrategies in its analysis. Multi-body strategies, however, only have signif-icant sensitivity when combined with detailed knowledge of the D mesondecay structure. Fortunately, the parameters associated with the specificmulti-body final states needed for these analyses can be extracted fromcorrelations within CLEO-c ψ (3770) data. a Here and subsequently, D will denote a D or ¯D
2. Determination of the CKM angle γ from B ± → DK ± The interference between B − → D K − and B − → ¯D K − decays, and equallybetween their CP conjugate states, provides a clean mechanism for theextraction of the CKM angle γ when the D and ¯D mesons decay to acommon final state, f D . Decay rates in these channels have the followingamplitude ratio A (B − → ¯D K − ) A (B − → D K − ) = r B e i ( δ B − γ ) , (1)which is a function of three quantities: the ratio of the amplitudes absolutemagnitudes r B , a CP invariant strong phase difference δ B , and the CKMweak phase γ . Generically, the amplitude for the complete decay B − → D( f D )K − , normalised to the favoured B → DK amplitude, is defined as A (B − → D( f D )K − ) A (B − → D K − ) = A D + r B e i ( δ B − γ ) A ¯ D , (2)where A D and A ¯ D represent the amplitudes for the D and ¯D decays,respectively. Due to colour suppression r B < .
13 @ 90% CL ; therefore,the interference is generally small. A variety of strategies exist, however,that attempt to resolve this and maximise the achievable sensitivity to γ .One such tactic is to consider multi-body final states of the D meson. ADS Formalism
Atwood, Dunietz and Soni (ADS) have suggested considering D decays tonon-CP eigenstates as a way of maximising sensitivity to γ . Final statessuch as K − π + , which may arise from either a Cabibbo favoured D decayor a doubly Cabibbo suppressed ¯D decay, can lead to large interferenceeffects and hence provide particular sensitivity to γ . This can be observedby considering the rates for the two possible B − processes:Γ(B − → (K − π + ) D K − ) ∝ r B r K πD ) + 2 r B r K πD cos (cid:0) δ B − δ K πD − γ (cid:1) , (3)Γ(B − → (K + π − ) D K − ) ∝ r B + ( r K πD ) + 2 r B r K πD cos (cid:0) δ B + δ K πD − γ (cid:1) , (4)where r K πD , [(61 . ± . × − ] , parameterises the relative suppressionbetween A D and A ¯ D , and δ K πD , [(22 +14 − ) ◦ ] , the relative strong phasedifference. ctober 25, 2018 20:9 Proceedings Trim Size: 9in x 6in powell-proc9x6 3 Since r B and r K πD are expected to be similar in magnitude, it can beseen that whilst Eq. (4) is the more suppressed of the two rates, it providesgreater sensitivity to γ as a result of the interference term appearing atleading order. Through considering the other two rates associated withthe B + decay, and combining this with information from decays to theCP-eigenstates K + K − and π + π − , an unambiguous determination of γ canbe made. The expected one-year sensitivity to γ from these six rates isestimated to be 8-10 ◦ at LHCb , depending on the value of δ K πD . Multi-body Extension to the ADS Method
A wealth of additional statistics can be gained from considering multi-bodydecays of the D meson. In the case of the ADS method, these are states in-volving a charged kaon and some ensemble of pions, such as D → K − π + π and D → K − π + π − π + . However, a complication comes from the fact thatthe multi-body D decay-amplitude is potentially different at any pointwithin the decay phase space, because of the contribution of intermediateresonances. It is shown in Ref. 10 how the rate equations for the two-bodyADS method should be modified for use with multi-body final states. Inthe case of the B − rates, for some inclusive final state f , Eq. (4) becomes:Γ(B − → ( ¯ f ) D K − ) ∝ ¯ A f + r B A f + 2 r B R f A f ¯ A f cos (cid:16) δ B + δ fD − γ (cid:17) , (5)where R f , the coherence factor, and δ fD , the average strong phase difference,are defined as: A = Z | A D ( x ) | d x , ¯ A = Z | A ¯D ( x ) | d x , (6) R f e iδ f D = R | A D ( x ) | | A ¯D ( x ) | e iζ ( x ) d x A f ¯ A f { R f ∈ R | ≤ R f ≤ } , (7)with x representing a point in multi-body phase space and ζ ( x ) the corre-sponding strong phase difference.
3. Determining R f and δ fD at CLEO-c Through exploiting the fact that meson pairs produced via quarkoniumresonances at e + e − machines are in quantum entangled states, it is possibleto obtain observables that are dependent on parameters associated withmulti-body decays. In particular, it has been shown in Ref. 10 that, double-tagged D ¯D rates measured at the ψ (3770) provide sensitivity to both thecoherence factor, R f , and the average strong phase difference, δ fD . Starting ctober 25, 2018 20:9 Proceedings Trim Size: 9in x 6in powell-proc9x64 with the anti-symmetric wavefunction of the ψ (3770) and then calculatingthe matrix element for the general case of two inclusive final states, F and G , the double-tagged rate is found to be proportional to:Γ( F | G ) ∝ A F ¯ A G + ¯ A F A G − R F R G A F ¯ A F A G ¯ A G cos( δ FD − δ GD ) . (8)From this, one finds three separate cases of interest for accessing boththe coherence factor and the average strong phase difference. These aresummarised in Table 1 below, in the instance of F = K πππ . Table 1. Double-tagged rates of interest and their dependence on the coherence factor, R K π ,and the average strong phase difference, δ K πD . The background subtracted yields from the818 pb − data sample are shown along with the corresponding result for each measurement. K ± π ∓ π + π − vs. Measurement 818 pb − Yield K ± π ∓ π + π − ( R K π ) = 0 . ± . ± .
07 30 ± R K π cos( δ K πD ) = − . ± . ± .
24 2,183 ± ± π ∓ R K π cos( δ KπD − δ K πD ) = − . ± . ± .
09 38 ± Event Selection
At present, only double-tagged samples for the determination of R K π and δ K πD have been analysed using CLEO-c’s complete ψ (3770) dataset, corre-sponding to an integrated luminosity of 818 pb − . To maximise statistics,a total of nine distinct CP tags are reconstructed against K ± π ∓ π + π − :K + K − , π + π − , K π , K ω/η ( π + π − π ), K π π , K φ , K η ( γγ ) andK η ′ ( π + π − η ). Backgrounds within these CP-tagged samples are typicallylow; in the range of ∼ ∼ Preliminary Results
From the background subtracted yields determined, central values havebeen calculated for R K π cos( δ K πD ) for each of the 9 CP-tags; for R K π cos( δ KπD − δ K πD ) using the K ± π ∓ π + π − vs. K ± π ∓ sample; and for( R K π ) using the observed number of K ± π ∓ π + π − vs. K ± π ∓ π + π − events.In addition, the results of the 9 separate CP-tags are used to form a com-bined result for R K π cos( δ K πD ), taking full account of correlations between ctober 25, 2018 20:9 Proceedings Trim Size: 9in x 6in powell-proc9x6 5 systematic uncertainties. The preliminary results are quoted in the 2 nd column of Table 1, where the first error is statistical and the second sys-tematic. The resulting constraints on the parameters R K π and δ K πD fromthese measurements are shown in Fig. 1. It is apparent, from Fig. 1, that π K3 R0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( d e g . ) D π K δ σ σ σ -1 CLEO-c preliminary 818 pb
Figure 1. The resulting limits on R K π and δ K πD at 1, 2 and 3 σ levels. the coherence across all phase space is low, reflecting the fact that manyout of phase resonances contribute to the K πππ final state. An inclusiveanalysis of this final state with the ADS analysis will therefore have lowsensitivity to the angle γ , although the structure of Eq. (5) makes it clearthat such an analysis will allow for a determination of the amplitude ratio r B , which is a very important auxiliary parameter in the γ measurement. References
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